In decision tree algorithms, selecting the optimal feature for splitting the data at each node is crucial for building an effective model. Two popular criteria used for this purpose are the **Gini Index** and **Entropy**. Both are measures of impurity or uncertainty, and they play a vital role in determining the quality of the splits in the decision tree. Although they serve similar purposes, they have distinct characteristics and implications for the decision tree’s structure and performance. This article explains the differences, similarities, and practical considerations of the Gini Index and Entropy.

**Gini Index definition**

The Gini Index, also known as Gini Impurity, measures the probability of incorrectly classifying a randomly chosen element if it was randomly labeled according to the distribution of labels in the dataset. It ranges between 0 (perfectly pure) and 0.5 (maximal impurity in a binary classification). The formula for the Gini Index for a node is:

\text{Gini}(S) = 1 - \sum_{i=1}^{n} p_i^2,where p_i is the probability of an element being classified into a particular class.

Key Characteristics of Gini Index:

- Simplicity: The Gini Index is computationally simpler and faster to calculate than Entropy, as it involves only squaring probabilities.
- Interpretation: A Gini Index of 0 indicates that all elements belong to a single class (pure node), while a value close to 0.5 suggests a more mixed or impure node.
- Bias: Gini tends to create biased splits, favoring features with more categories or those with higher probability classes.

**Entropy ****definition**

Entropy, derived from information theory, quantifies the uncertainty or disorder in a dataset. It measures the expected amount of information required to classify a new example. The formula for Entropy for a node is:

H(S) = - \sum_{i=1}^{n} p_i \log_2 p_i,where p_i is the probability of an element being classified into a particular class.

Key Characteristics of Entropy:

- Complexity: Entropy is slightly more complex to calculate compared to Gini, due to the logarithmic function.
- Interpretation: Like the Gini Index, an Entropy value of 0 indicates a pure node, while higher values suggest more impurity.
- Sensitivity: Entropy is more sensitive to changes in the data distribution, which can make it a more nuanced measure in certain contexts.

In a decision tree, the feature with the highest Information Gain or the lowest Gini Impurity (measures of impurity or disorder) is typically chosen as the root node. The Gini Index ranges between 0 and 1, while Entropy can exceed 1.

Consider a set of classes “Yes” and “No” with a total of 20 members. Then:

– If the number of “Yes” equals the number of “No”:

\text{Entropy}(S) = -\sum_{j} p_j \log_2 p_j = -\left(\frac{1}{2} \log_2 \frac{1}{2} + \frac{1}{2} \log_2 \frac{1}{2}\right) = -\left(-\frac{1}{2} \cdot 1\right) = 1

– If all are “Yes” or all are “No”:

\text{Entropy}(S) = -\sum_{j} p_j \log_2 p_j = -1 \cdot \log_2 1 = -1 \cdot 0 = 0

– For such binary division, it does not matter which one is more:

- \text{Entropy}(S) = -\left(0.75 \log_2 0.75 + 0.25 \log_2 0.25\right)= 0.811
- \text{Entropy}(S) = -\left(0.25 \log_2 0.25 + 0.75 \log_2 0.75\right) = 0.811

### Entropy values vs. Gini values by plot

To illustrate the relationship between Entropy and Gini Index values, let’s consider an example where we have a binary classification problem with two events: $X=0$ and $X=1$. We will plot the Entropy and Gini values as the proportion of $X=1$ varies from 0 to 1.

The Entropy curve starts at 0 when $p_{1}$ is either 0 or 1 and reaches its peak of 1 when $p_{1}$ is 0.5. This reflects the uncertainty or disorder within the dataset, with higher Entropy values indicating greater uncertainty. Similarly, the Gini Index also starts at 0 when $p_{1}$ is 0 or 1 and peaks at 0.5 when $p_{1}$ is 0.5. It measures the impurity or diversity of the dataset, with higher Gini values indicating greater impurity.

**If we multiply the Gini values by 2**, the resulting plot aligns closely with the Entropy plot. This is because multiplying an objective function by a positive number does not affect the optimal solution—in this case, selecting the best feature to split the data. Thus, whether you use the Entropy or Gini Index formula does not impact the accuracy of the model. However, the choice between them can affect computational speed.